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The mathematical method explained

A full academic research paper, for publication in a peer-reviewed journal,
is currently in preparation. That paper will contain all the
details of the alt-3 method, along with the mathematical justification for its
use to produce routinely better league-table rankings.

For now, we will show here the key mathematical formulae behind the method —
for those who enjoy such things!

The mathematical model

The essence of alt-3 is to determine a numeric, positive-valued
strength, ,
for each team in the league, on the basis of the team’s
match results to date within the current season. The scale on which
team strength is measured is such that an ‘average’ team has strength
.

The strengths are formally parameters in a statistical model —
a model which
specifies the probability of every possible match outcome for the whole season.

Specifically, when home team plays away team , the three possible
match outcomes (that is to say: wins, wins, or a draw)
are taken to have the following probabilities:

The model depends on two further numerical quantities, in addition
to the team-specific strengths . These are:

: the ‘home advantage’ parameter.
In football leagues, this
will be a number greater than 1. For example, a value of
would mean that a team’s strength is enhanced by 10%
when playing at home.
The alt-3 model uses the same home-advantage effect for
every team within a league.
(A more elaborate model, with home advantage allowed to
differ between teams, would not work at all well for our purpose,
since it would confound the interpretation of league tables based upon it.)

: the ‘prevalence of draws’ parameter.
The specific interpretation
of is that the probability of a draw between two average-strength
teams (that is, teams and with ),
in a hypothetical match where neither team has home advantage, is
. Thus, for example, the value would
imply that the probability of a draw in such a hypothetical match between
average teams is .

The home-advantage parameter and the draw-prevalence parameter
are taken to be structural quantities, meaning that they
can reasonably be assumed to be constant across different seasons of any specific
league. This allows the values of and to be determined
from historical data, for any given league.

There is no reason to suppose that different leagues should share the same
home-advantage or draw-prevalence properties. Indeed, there are some marked
differences between the major leagues in Europe, in regard to the
numerical values of and that fit well to historical data.

Isn’t that just a generalized Bradley-Terry model?

Yes, it is!

(But that is a nerdy question, of course.
Don’t worry if you have never heard of the
Bradley-Terry model before. The main point of this part of the page
is to say briefly where that strange-looking 1/3 power comes from,
in the formulae above — and why it is important.)

The Bradley-Terry model, and variants of it, are well established in the
ranking of sports players and teams. A well-known example is the
Krach
method that is used in US college hockey (i.e., ice hockey).
And the
Elo rating
method, well known especially for its use to rank
chess players, is also closely related to the Bradley-Terry model.

The probabilities shown above are in fact very similar to the model introduced in
Davidson and Beaver (1977),
which extends the Bradley-Terry model to handle
draws (ties) and a home-advantage effect (or ‘order effect’). The only material
difference is the power 1/3 that appears in our formulae above, which
determines how the probability of a draw relates to relative team strengths;
the Davidson-Beaver model has power 1/2 instead.

The reason for the power 1/3 is simple: in modern football leagues, three
league points are awarded for a win, and 1 for a draw. It is long established
that the Davidson-Beaver model, with its 1/2 power, would always
reproduce correctly the
final league standings in a full round-robin tournament, provided that the
points system is ‘2–1–0’, i.e., two points for a win and one for a draw.
The appropriate modification for a 3–1–0 points system is to use power 1/3
instead of 1/2.

The use of power 1/3 is crucial, for a coherent ranking that
takes into account schedule strength in football leagues. The use of power 1/3
ensures that whenever schedule-strength differences are absent, the model-based
‘retrodictive’ ranking agrees exactly (as it surely must!) with the official
league table. This is particularly important, of course, at the end
of the season, when the full double round-robin tournament structure
exactly eliminates all schedule-strength differences.

How are the team strengths determined?

The strengths are determined to make the above probabilities
fit as well as possible to the actual match results
seen already.

The standard statistical
method for this is the method of maximum likelihood. This
method finds the unique values of strengths that yield exact agreement
between:

the actual league points totals to date; and

the expected league points, as derived from the above model, for the
matches already played.

(It is the choice of power 1/3 in the model that ensures that the method of
maximum likelihood has this exact relationship with league points.)

Expected points per match

The assumed probabilities shown above,
for the three
possible match outcomes ‘ beats ’, ‘ and draw’ and
‘ beats ’, can be used straightforwardly to obtain team
’s expected
points (, say) in the home match against .
With 3 points for a win and 1 for a draw:

Let denote the average of all such
and values,
i.e., averaged over all of the matches played by in a whole
double-round-robin season. That is, is the expected (or projected)
points per match for team (both home and away)
over the entire season — based on all of the
current
team-strength values , and on the structural constants
(home advantage) and (draw prevalence).

Then an easily established (and fairly obvious?)
fact is that the values of are monotonically
related to the strengths . In other words, ordering the teams by
their values of is equivalent to ordering them by the the values
of .

The projected, season-long points per match rate is precisely
the Rate column that is used to order the alt-3 table.

Expression in terms of ‘schedule strength’ and ‘effective matches played’

For any match in which plays , with at home say, we define
the schedule strength of that match for team to be

In this way, an ‘average’ match for has schedule-strength zero.
Tougher than average matches have positive-valued schedule strength; while
easier matches than average have negative schedule strength.

The total schedule strength of all of ’s matches over the whole season
is then (by design) exactly zero.

At any point during the season, we now define the
effective number of matches played by team , by subtracting ’s total
schedule strength (to date) from the number of actual matches already
played by :

ePld = Pld − (schedule strength)

And finally we will see the reason for these definitions. The beauty of
ranking via the alt-3 method is that, for ’s matches already played,
the total league points accumulated by (denoted by Pts
in the league table) is exactly the same as the sum of the model-expected
points — i.e., precisely the same sum that
appears in ’s total schedule strength.

So we can now write

ePld = Pld − (Pld − Pts / )
= Pts / ,

and hence the projected whole-season points per match for
can be usefully re-expressed as

= Pts / ePld.

This provides a cleaner interpretation for that all-important
Rate column in the alt-3 table
— an interpretation that completely avoids any (potentially misleading)
predictive connotations. The
Rate for any given team is best thought of as
league points gained per
effective match played.

The difference between the actual and effective matches played is the current
(cumulative total)
schedule strength for team . This is the key to understanding the positions
of teams on the alt-3 table. For that reason, a graph of
every team’s match-by-match schedule strength is provided along
with the alt-3 table itself: just click on any team’s name
to see the full (current) schedule-strength chart.